Finely grate the zest of 3 of the lemons into a bowl. Halve all the lemons, then use a fork to squeeze out all the juice and pulp into the bowl with the zest - don’t worry about the seeds. Crack the eggs into a separate bowl, then whisk in the caster sugar until completely combined. Whisk in the cream and lemon juice mix, then set aside.
Place a 23cm tart ring or loose-bottom flan tin on a baking sheet lined with greaseproof paper. On a surface dusted with icing sugar, roll out the pastry to the thickness of a £1 coin. Lift onto a rolling pin, then drape over the tart ring or flan tin, leaving the excess hanging over the edge. Carefully press the pastry into the sides with a small ball of pastry. Put in fridge or freezer for 20 mins to allow pastry to chill.
While the pastry is chilling, heat oven to 200C/fan 180C/gas 6. Prick the base of the pastry case, line with greaseproof paper and baking beans, then bake for 15 mins. Remove the paper and beans, then bake the case for 5-10 mins more until it is biscuity brown and crisp. Lower oven to 150C/fan 130C/gas 2.
Use a ladle to push the lemon custard through a sieve into a bowl, pressing down on the pulp to extract as much juice as possible. Skim the bubbles from the surface of the custard, then pour the custard into a jug. Pull the tart out of the oven slightly, then pour in the custard so it comes to the top. You may have some left over. Push tart back in, then bake for 35-40 mins until the top forms a light crust and the custard is just set. Leave to cool, then chill.
While the tart is cooling, make the berry salad. Place a large non-stick frying pan over a medium heat and scatter the icing sugar over the base. Heat until the sugar dissolves and caramelises. Throw in the berries, toss with the sugar, then splash in the sherry vinegar. Bubble for a min until everything is just starting to soften, then tip onto a plate and leave to cool slightly.
Use a serrated knife to trim away the pastry and make a neat edge. Dust the tart lightly with icing sugar, then use a blowtorch to caramelise the top. Leave the top to set, then repeat the process so that you get an extra-thick layer of caramel. Stir the mint through the berries. Serve the tart in slices with a bowl of the berry salad on the side.